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But then gave myself a terrible headache trying to just substitute in values and use trial and error to get them to balance. Did I go about it completely the wrong way?? Or is there another step to eliminate the possibilities once you've reached the above ??

Ok, probably some algebraic way to do this, but you can do it via deduction. I'll try to include all details on all steps and even harder, try to put it in a way that makes sense

Spoiler for explanation

1. t = 1 (has to, because S + S + carryover can not add up to a # greater than 20)
2. E = 8 (not 0, because i would have to equal 1 or 0, which would not be true since 0 and 1 are already represented - assumption being made that it's not 9, which is possible, and this is generally a mistake, but it worked here - if would have been a mistake, realization would have probably occurred quickly)
3. i = 5 (assume no carryover from ones column and we test this, hence gets nasty here - we have to look at ones column - by process of elimination - n < 5; n = 4, or x would equal 1 , but t = 1 already - n does not equal 3, or else x would equal 1 or 2, 1 already explained, with 2, y would equal 8, which is E, so that's out. N must equal 2.)
4. n =2
5. v = 7 (not 6 or 9 because S would have to equal E; we know a 2 is carried over to next column, so anything less than 6 would not work)
6. S = 6 (only option b/c 3rd column already solved)
7. w = 3
8. x = 0
y = 4